Reply to Discussion: “On the Thermodynamical Admissibility of the Triphasic Theory of Charged Hydrated Tissues” (Mow, V. C., Lai, W. M., Setton, L. A., Gu, W., Yao, H., and Lu, X. L., 2009, ASME J. Biomech. Eng., 131, p. 095501)

First, we warmly thank the authors of the above cited discussion for their genuine interest in the paper published by us in Ref. 1, as well as for their detailed comments on this paper. We address each of their points in the order in which they appear in the discussion (2).

• 1

  We are very happy that Mow et al. hold the derivation of constitutive restrictions from the second law of thermodynamics as the reference. We indeed agree that in the first half of the paper (2) the second law of thermodynamics is used to formulate restrictions for the constitutive behavior of the triphasic medium. Unfortunately, Lai et al. do not give expressions for the Helmholtz free energies $As$⁠, $Af$⁠, $A+$⁠, and $A−$⁠, in order to derive Eqs. (46), (60), and (61) from the thermodynamic restrictions (26) and (27).

• 2

  We highly appreciate that Mow et al. agree that the electrochemical potentials are strain-dependent. This is exactly the point we emphasized in Ref. 1. The footnote on p. 252 of Ref. 3 states that the electrochemical potential depends on solid dilatation, but also states that this dependence is negligible. We demonstrate in Ref. 1 that this contribution is not negligible.

• 3

  No comments.

• 4
  Okay. The definition that Lai et al. (3) handled for the fluid volume fraction is Eulerian (current fluid volume/current mixture volume, while Huyghe et al. (1) handled the Lagrangian definition (current fluid volume/initial mixture volume), which is not consistent with Ref. 3. So the notation of $ϕw$ in Ref. 1 is not consistent with the notation $ϕw$ in Ref. 3. All occurrences of $ϕw$ in Ref. 1 should be replaced by $(1+E)ϕw$⁠. Indeed,

  $VwV0=VV0VwV=(1+E)ϕw$

  1

  in which $Vw$ is the current fluid volume, $V0$ is the initial mixture volume, and $V$ is the current mixture volume. However, this alters neither the further elaboration of the mathematics, nor the conclusions of the paper. We heartfully thank Mow et al. (2) for pointing out this inconsistency in the paper (1).
• 5

  The value of free energy production is 20% of the elastic energy stored in the sample. This is not a small second order effect. Swelling materials are capable of performing work. For instance, a swelling material can raise loads. This work performed onto the environment is done at the expense of the free energy contained within the material. If a triphasic model is equipped with a chemical expansion stress that doubles its swelling propensity compared with a Donnan model, then its capacity to perform work onto the environment is doubled as well. This extra work should be performed at the expense of extra loss of free energy within the material. Otherwise, the chemical expansion stress is like a car running without petrol. The strain-dependency of the (electro)chemical potential is the term in the equations that ensures that this extra loss of free energy is accounted for, and is not a second order term.

• 6

  No comments.

• 7

  We very much support the leading role that Mow et al. (2) attribute to experimental data. In fact, the chemical expansion stress was introduced in the first place on the basis of experiments (Fig. 8 in Ref. 3). We realize fully that after 1999 the chemical expansion stress disappeared from their papers. To our knowledge, this choice is hardly justified in any of the papers. The lack of experimental data for solute (electro)chemical potential in cartilage is no excuse to neglect the strain-dependence of the solute (electro)chemical potentials. The constitutive restrictions from the second law tell with utmost precision what the strain-dependence is, given an expression for the chemical expansion stress. Mow et al. (2) object against the expression “distrust results obtained from the triphasic theory” in our technical brief. The intention of the technical brief is not at all to crack down on any of their work, which we recognize as a cornerstone in the development of mechano-electrochemical theory of cartilaginous tissues, but rather to foster a discussion between scientists that will inevitably lead to even greater heights of fulfillment of the science of cartilaginous tissues.

In conclusion, the authors thank Mow et al. (2) for their detailed expert discussion of the paper entitled “On the Thermodynamic Admissibility of the Triphasic Theory of Charged Hydrated Tissues” (1), but unfortunately, they cannot support all of their conclusions.